oh, yes, they are zeners but everything still holds in my explanation, theyre just conducting in breakdown. oops.
as for it being convoluted, perhaps for a novice, but thats the point of the problem: learning (zener) diodes, not ohms law or KVL.
once you know how zeners work you can figure the voltage at any point in that circuit just by looking at it, and the current through any of the resistors almost as fast. though it takes far more text to describe an intuition-based solution.
your method checks to see if the zeners are in breakdown mode, which is the correct way to do it. but once you know that they are, you have to re-calculate the current in the resistors, hence where my solution comes in.